A132044 Triangle T(n,k) = binomial(n, k) - 1 with T(n,0) = T(n,n) = 1, read by rows.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 3, 1, 1, 4, 9, 9, 4, 1, 1, 5, 14, 19, 14, 5, 1, 1, 6, 20, 34, 34, 20, 6, 1, 1, 7, 27, 55, 69, 55, 27, 7, 1, 1, 8, 35, 83, 125, 125, 83, 35, 8, 1, 1, 9, 44, 119, 209, 251, 209, 119, 44, 9, 1

Offset: 0

Author

Gary W. Adamson, Aug 08 2007

Keywords

• nonn
• tabl

Comments

Row sums = A132045: (1, 2, 3, 6, 13, 28, 59, ...).

The triangle sequences having the form T(n,k,q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 have the row sums Sum_{k=0..n} T(n,k,q) = 2^(n-2)*q^n + 2^n - (n-1) - (5/4)*[n=0] -(q/2)*[n=1]. - G. C. Greubel, Feb 12 2021

Examples

			First few rows of the triangle:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  2,  1;
  1, 3,  5,  3,  1;
  1, 4,  9,  9,  4,  1;
  1, 5, 14, 19, 14,  5,  1;
  1, 6, 20, 34, 34, 20,  6, 1;
  1, 7, 27, 55, 69, 55, 27, 7, 1;
		

Links

• G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Crossrefs

Cf. A007318, A103451, A132045.

Cf. this sequence (q=0), A173075 (q=1), A173046 (q=2), A173047 (q=3).

Programs

Magma

T:= func< n,k,q | k eq 0 or k eq n select 1 else Binomial(n,k) + q^n*Binomial(n-2,k-1) -1 >;
[T(n,k,0): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 12 2021

Mathematica

T[n_, k_]:= If[k==0 || k==n, 1, Binomial[n, k] - 1];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* Roger L. Bagula, Feb 08 2010 *)

Sage

def T(n,k,q): return 1 if (k==0 or k==n) else binomial(n,k) + q^n*binomial(n-2,k-1) -1
flatten([[T(n,k,0) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 12 2021

Formula

T(n, k) = A007318(n,k) + A103451(n,k) - A000012(n,k), an infinite lower triangular matrix.

T(n, k) = binomial(n, k) - 1, with T(n,0) = T(n,n) = 1. - Roger L. Bagula, Feb 08 2010

From G. C. Greubel, Feb 12 2021: (Start)

T(n, k, q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 with T(n, 0) = T(n, n) = 1 and q = 0.

Sum_{k=0..n} T(n, k, 0) = 2^n - (n-1) - [n=0]. (End)